PARAMETRIC VIBRATIONS OF ORTHOTROPIC PLATES WITH COMPLEX SHAPE L. V. Kurpa * and O. S. Mazur ** The paper proposes a method to study the parametric vibrations of orthotropic plates with complex shape. The method is based on the R-function theory and variational methods. Dynamic-instability domains and amplitude–frequency responses for plates with complex geometry and different types of boundary conditions are plotted Keywords: parametric vibrations, dynamic instability, R-function theory, variational methods, orthotropic plate Introduction. Studying the vibrations of plates and shells under periodical loading is of great practical importance because certain combinations of load parameters and natural frequency may cause a parametric resonance, leading to loss of integrity of the structure. This issue was addressed in many publications [2, 15–17, etc.]. To identify domains in which the amplitude of vibrations intensively increases, the system of equations is usually [2, 4, 9, 15, 18] reduced to the Mathieu–Hill equation or to a system of such equations whose properties are well understood. As shown in [3, 10, 11], with certain ratios among its coefficients, the Mathieu–Hill equation has solutions increasing without limit. The associated values of the parameters define the dynamic-instability domains (DIDs). One of the primary tasks in studying the parametric vibrations of plates is to plot their resonant amplitude–frequency responses (AFRs). To this end, the small-parameter method, the method of trigonometric series, the asymptotic Krylov–Bogolyubov method, etc., were used in [2–4, 15]. This made it possible to analyze the solutions for stability for the load parameters corresponding to the principal DID. Note that the methods used in the publications cited above allow studying the vibrations of plates and shells with a canonical planform (rectangle, circle). The present paper proposes a numerical analytic method based on the classical ideas underlying the nonlinear dynamics of plates and shells [2, 4], the R-function theory [14], and variational methods [13]. By combining these methods, we have developed a universal method for studying the parametric vibrations of orthotropic plates with different shapes and different types of boundary conditions. 1. Problem Formulation. Consider an orthotropic plate of constant thickness h under a periodical load p(t) acting in its plane. We will study the nonlinear vibrations of the plate assuming that there are no elastic waves; therefore, the inertial forces along the OX- and OY-axes can be neglected. The mathematical formulation of the problem based on the Kirchhoff–Love hypotheses and the assumption that the deformation of the plate is geometrically nonlinear is reduced to the following system of differential equations [1, 5]: ¶ ¶ + ¶ ¶ = N x T y x 0, ¶ ¶ + ¶ ¶ = T x N y y 0, (1.1) D w x D w y D w x y w x N w xy T x 1 4 4 2 4 4 3 4 2 2 2 2 2 2 2 ¶ ¶ + ¶ ¶ + ¶ ¶ = ¶ ¶ + ¶ ¶¶ + ¶ 2 2 2 2 w y N w t h w t y ¶ - ¶ ¶ - ¶ ¶ e r , (1.2) International Applied Mechanics, Vol. 46, No. 4, 2010 438 1063-7095/10/4604-0438 ©2010 Springer Science+Business Media, Inc. National Technical University “KhPI,” 21 Frunze St., Kharkiv, Ukraine, e-mail: * kurpa@kpi.kharkov.ua, ** mazur.olga@mail.ru. Translated from Prikladnaya Mekhanika, Vol. 46, No. 4, pp. 83–95, April 2010. Original article submitted December 24, 2008.